Question

plot the point with the given polar coordinates.

1. $(3,150^{circ})$
2. $(2,-255^{circ})$
3. $(3,\frac{11\pi}{6})$
4. $(-3,\frac{23\pi}{12})$

Explanation:

Step1: Understand polar coordinates

Polar coordinates \((r,\theta)\) where \(r\) is the distance from the origin and \(\theta\) is the angle. For positive \(r\), we move \(r\) units along the terminal side of \(\theta\). For negative \(r\), we move \(|r|\) units in the opposite direction of the terminal side of \(\theta\).

Step2: Plot \((3,150^{\circ})\)

• Start at the origin.
• Measure an angle of \(150^{\circ}\) (in standard position, counter - clockwise from the positive \(x\) - axis).
• Move 3 units along the terminal side of the \(150^{\circ}\) angle.

Step3: Plot \((2, - 255^{\circ})\)

• Since the angle is negative, measure \(255^{\circ}\) clockwise from the positive \(x\) - axis.
• \( - 255^{\circ}+360^{\circ}=105^{\circ}\), so it is equivalent to an angle of \(105^{\circ}\) in the counter - clockwise direction (but we use the clockwise method for plotting negative angles).
• Move 2 units along the terminal side of the \(-255^{\circ}\) (or equivalent terminal side) angle.

Step4: Plot \((3,\frac{11\pi}{6})\)

• Convert \(\frac{11\pi}{6}\) to degrees: \(\frac{11\pi}{6}\times\frac{180^{\circ}}{\pi}=330^{\circ}\).
• Start at the origin, measure an angle of \(330^{\circ}\) (counter - clockwise from the positive \(x\) - axis).
• Move 3 units along the terminal side of the \(330^{\circ}\) (\(\frac{11\pi}{6}\)) angle.

Step5: Plot \((-3,\frac{23\pi}{12})\)

• First, \(\frac{23\pi}{12}=23\times15^{\circ}=345^{\circ}\).
• Since \(r=-3\), instead of moving along the terminal side of \(\frac{23\pi}{12}\) (\(345^{\circ}\)), we move 3 units in the opposite direction of the terminal side of \(\frac{23\pi}{12}\). The opposite direction of \(345^{\circ}\) is \(345^{\circ}-180^{\circ}=165^{\circ}\).

Answer:

1. For \((3,150^{\circ})\): Locate the \(150^{\circ}\) angle (in the second quadrant, \(30^{\circ}\) above the negative \(x\) - axis) and mark a point 3 units from the origin along that terminal side.
2. For \((2,-255^{\circ})\): Locate the \(105^{\circ}\) (equivalent terminal side when considering positive angles for plotting) or plot by moving 2 units along the terminal side of the \(-255^{\circ}\) (clockwise \(255^{\circ}\) from the positive \(x\) - axis, which is in the second quadrant).
3. For \((3,\frac{11\pi}{6})\): Locate the \(330^{\circ}\) (\(\frac{11\pi}{6}\)) angle (in the fourth quadrant, \(30^{\circ}\) below the positive \(x\) - axis) and mark a point 3 units from the origin along that terminal side.
4. For \((-3,\frac{23\pi}{12})\): Locate the direction opposite to the \(345^{\circ}\) (\(\frac{23\pi}{12}\)) angle (which is \(165^{\circ}\)) and mark a point 3 units from the origin along that opposite - direction terminal side.